Mesoscale variable velocity

Define phase space of mesoscale variables needed to describe a particle (velocity, volume, etc.)... 

As discussed in Chapter 2, the one-particle NDF does not usually provide a complete description of the microscale system. For example, a microscale system containing N particles would be completely described by an A-particle NDF. This is because the mesoscale variable in any one particle can, in principle, be influenced by the mesoscale variables in all N particles. Or, in other words, the N sets of mesoscale variables can be correlated with each other. For example, a system of particles interacting through binary collisions exhibits correlations between the velocities of the two particles before and after a collision. Thus, the time evolution of the one-particle NDF for velocity will involve the two-particle NDF due to the collisions. In the mesoscale modeling approach, the primary physical modeling step involves the approximation of the A-particle NDF (i.e. the exact microscale model) by a functional of the one-particle NDF. A typical example is the closure of the colli-sionterm (see Chapter 6) by approximating the two-particle NDF by the product of two one-particle NDFs. 

The transport equations appearing in macroscale models can be derived from the kinetic equation using the definition of the moment of interest. For example, if the moment of interest is the disperse-phase volume fraction, then it suffices to integrate over the mesoscale variables. (See Section 4.3 for a detailed discussion of this process.) Using the velocity-distribution function from Section 1.2.2 as an example, this process yields... 

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

The second example is the case in which the disperse-phase velocity is equal to the conditional expected disperse-phase velocity given the other mesoscale variables Vp = (Upl p, Vf, f). In this case, fluctuations in the disperse phase are slaved to the fluid-phase fluctuations and can depend on the particle internal coordinates (e.g. particle size). The NDF in this case is n(Vp, p, Vf, f) = n( p, Vf, f)(5(Vp - [Pg.131]

A popular method for closing a system of moment-transport equations is to assume a functional form for the NDF in terms of the mesoscale variables. Preferably, the parameters of the functional form can be written in closed form in terms of a few lower-order moments. It is then possible to solve only the transport equations for the lower-order moments which are needed in order to determine the parameters in the presumed NDF. The functional form of the NDF is then known, and can be used to evaluate the integrals appearing in the moment-transport equations. As an example, consider a case in which the velocity NDF is assumed to be Gaussian ... 

In order to account for variable particle numbers, we generalize the collision term iSi to include changes in IVp due to nucleation, aggregation, and breakage. These processes will also require models in order to close Eq. (4.39). This equation can be compared with Eq. (2.16) on page 37, and it can be observed that they have the same general form. However, it is now clear that the GPBE cannot be solved until mesoscale closures are provided for the conditional phase-space velocities Afp)i, (Ap)i, (Gp)i, source term 5i. Note that we have dropped the superscript on the conditional phase-space velocities in Eq. (4.39). Formally, this implies that the definition of (for example) [Pg.113]

Atmospheric mesoscale models are based on a set of conservation equations for velocity, heat, density, water, and other trace atmospheric gases and aerosols. The equation of state used in these equations is the ideal law. The conservation-of-velocity equation is derived from Newton s second law of motion (F = ma) as applied to the rotating earth. The conservation-of-heat equation is derived from the first law of thermodynamics. The remaining conservation equations are written as a change in an atmospheric variable (e.g., water) in a Lagrangian framework where sources and sinks are identified. 

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